The compensation is believed to be the biggest in British history and comes after an 18-year battle by the parents of Zach Petrou.
Zach was a healthy seven-month-old baby when he was admitted to Stepping Hill hospital in Stockport with the viral infection croup in 1994.
He had emergency treatment after his condition deteriorated. before being transferred to a second hospital. When he was taken off a ventilator there, Zach was found to be severely brain damaged.
His parents Margaret and Andros, from Tameside, Greater Manchester, believe he was deprived of oxygen during his treatment at Stepping Hill and that caused his injuries.
They were determined to fight for their son because of his round-the-clock care needs. And finally health chiefs have agreed the settlement.
The Manchester Evening News reported that Stepping Hill had not admitted liability but had agreed to pay an eight figure sum to Zach's family, to help pay for his care.
Mum Margaret, 54, said: "We're just very relieved. This result does bring us some level of comfort. One of our biggest concerns was what would happen to Zach when we were no longer here to care for him. We can now concentrate on his care and get him the best of what he needs."
Zach, now 19, functions at the level of a six to eight month old baby and needs round-the-clock care from at least two full time carers. He is expected to live beyond the age of 70.
Dad Andros, 50, said: "We have been left devastated by what happened to Zach. When we realised how badly brain damaged he was we were inconsolable. His future was taken away from him at that point.
"And the hardest thing to come to terms with has been the fact that we have always believed that what happened to Zach was entirely avoidable.
"Zach went into the hospital a healthy baby boy but was given back to us severely brain damaged. We are still grieving for the baby boy we lost that day.
"The hospital refused to accept responsibility. We were forced to go down the legal route just to get justice for our son. We still believe that what happened to Zach should never have happened and we hope that it never happens to another family again."
A spokesman for NHS North West said: "We are pleased that the parties involved in this case have been able to work together to agree a liability settlement which has been approved by the court.
"The Strategic Health Authority would like to pay tribute to the loving and devoted care that Zach's family have provided over the years and also would like to take this opportunity to wish Zach and his family well for the future."